Tag: angle made by a chord and a tangent

The range of values of $\lambda$ for which the circles $ { x }^{ 2 }+{ y }^{ 2 }=4$ and ${ x }^{ 2 }+{ y }^{ 2 }-2\lambda y+5=0$ have two common tangents only is-

The range of values of x for which the circles ${ x }^{ 2 }+{ y }^{ 2 }=4$ and$ { x }^{ 2 }+{ y }^{ 2 }+2xy+5=0\quad$ have two on tangents only is= 

Intercept of a tangent between two parallel tangents to a circle subtends a right angle at the centre.

In the given figure, $AD\ and AE$ are the tangents to a circle with centre $O\ and BC$ touches the circle at $F$. If $AE=5\ cm$ then perimeter of $\triangle ABC$ is 

$\overline { M N }$ and $\overline { M Q }$ are two tangents from a point $M$ to a circle with centre $0$ If $m \angle N O Q = 120 ^ { \circ } ,$ then ?

If $\triangle ABC$ is isoscles with $AB=AC$ and $C(O,r)$ is the incircle of the of the $\triangle BAC=30^{o}$. The tangent at $C$ intersects $AB$ at a point $D$, then $L$ trisects $BC$.

The chord of contact of the pair of tangents to the circle $x^2+y^2=1$ drawn from any point on the line $2x+y=4$ passes through a fixed point. 

From a point $P$ which is at a distance of $13$ cm from the centre $O$ of a circle of radius $5$ cm, the pair of tangents $PQ$ and $PR$ to the circle are drawn. Then the area of the quadrilateral $PQOR$ is:

Circles ${ C } _{ 1 },{ C } _{ 2 },{ C } _{ 3 }$ have their centres at $\left( 0,0 \right) ,\left( 12,0 \right) ,\left( 24,0 \right) $ and have radii $1,2$ and $4$ respectively. Line ${t} _{1}$ is a common internal tangent to ${C} _{1}$ and ${C} _{2}$ and has a positive slope and line ${t} _{2}$ is a common internal tangent to ${C} _{2}$ and ${C} _{3}$ and has a negative slope. Given that lines ${t} _{1}$ and ${t} _{2}$ intersect at $(x,y)$ and that $x=p-q\surd r$, where $p,q$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.

For the two circles ${ x }^{ 2 }+{ y }^{ 2 }=16$ and ${ x }^{ 2 }+{ y }^{ 2 }-2y=0$ there is/are